# Fundamental groups of symplectic leaves

Let $X = \operatorname{Spec}(R)$ be a conical Poisson variety over $\mathbb{C}$. This means that $R$ is a non-negatively graded Poisson algebra over $\mathbb{C}$ with $R_0 = \mathbb{C}$ and that the Poisson bracket is homogeneous of negative degree. Assume that $X$ admits a conical symplectic resolution. Examples to keep in mind are:

• the nilpotent cone in a simple Lie algebra
• a Slodowy slice to a nilpotent orbit inside the nilpotent cone
• the symmetric scheme of $n$ points on a Kleinian singularity
• a hypertoric variety
• a quiver variety.

Let $S$ be a symplectic leaf of $X$.

Q: Is it possible for the fundamental group of $S$ to be infinite?

In the first four examples listed above, the fundamental group of every symplectic leaf is finite. I don't know whether or not this property holds for quiver varieties, or for other examples.

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If you consider quiver varieties of affine types, they are moduli spaces of instantons on ALE spaces. If you replace the base space by $\mathbb R^4$ and assume rank is $2$, then the finiteness follows from Atiyah-Jones conjecture for rank $2$. The Atiyah-Jones conjecture actually says much stronger: there is a homotopy equivalence between moduli spaces and the infinite dimensional space of all connections up to dimension explicitly given by the instanton number. The conjecture was proved by Boyer et al.